4 semester


Course title

Differential equations


Year of study,

study programme


1-31 03 01-03 Mathematics (Economic Activities)


Semester of study



Number of credits




Gromak Valery Ivanovich


Course Objective

Formation of students’ skills and abilities in conducting mathematical research on the basis of the theory of differential equations, learning the basic analytical, qualitative and asymptotic methods of the theory of differential equations, constructing and analyzing basic mathematical models on the basis of the theory of differential equations.

As a result of the training, the student must


– basic concepts and methods of general theories of linear differential equations, existence and uniqueness theorems;

– basic methods of integrating linear differential equations and systems, Euler’s method, elimination method, matrix method;

– formulation of the Cauchy problem and boundary value problems;

– the basic concepts and methods of the theory of stability by Lyapunov.

be able to:

– apply methods of general theories of linear differential equations for solving basic types of linear differential equations and systems;

– to set initial and boundary-value problems, to solve questions of existence and uniqueness of the solution of initial and boundary value problems

– to build phase portraits of the simplest autonomous systems on the plane.

– apply the main theorems of the second Lyapunov method for solving the problems of stability of motion.



Algebra and Number Teory. Mathematical analysis. Geometry.


Course content

Linear differential equations of the n-th order. Properties of solutions of linear homogeneous differential equations of n-th order. The determinant of Vronsky. A fundamental system of solutions. Formula Ostrogradsky – Liouville. Method of variation of arbitrary constants for inhomogeneous linear differential equations. The Cauchy method. Linear homogeneous equations of the n-th order with constant coefficients. Euler’s method. Linear inhomogeneous equations of the n-th order with constant coefficients and the right-hand side of a special form. Second-order linear equations and oscillatory phenomena. Sturm’s theorems on the zeros of solutions. The concept of boundary value problems. Linear differential equations with holomorphic coefficients. Generalized power series. Integration of linear differential equations by means of power and generalized power series. The Airy and Bessel equation. Bessel functions.

Linear differential systems. Properties of solutions Linear independence of vector functions. Formula Ostrogradsky-Liouville. A fundamental system of solutions. The fundamental matrix. Method of variation of arbitrary constants for inhomogeneous linear systems. Exponential function of the matrix argument. The Lappo-Danilevsky theorem. Matrix method of integrating a linear homogeneous system with constant coefficients. The structure of the fundamental matrix. Euler’s method. Linear systems with periodic coefficients. Reduced systems. Solution of an inhomogeneous system with a right-hand side of a special form.

Autonomous systems of differential equations. Autonomous systems. Phase portraits of a linear autonomous system of two equations. Special points: node, saddle, focus, center. The concept of a limit cycle. The problem of focus and focus.

Lyapunov stability of solutions of differential equations. Functions of Lyapunov. Lyapunov’s theorems on stability and asymptotic stability. A criterion for the asymptotic stability of the zero solution of linear autonomous systems and the nth order equation. Lyapunov’s theorem on stability in the first approximation.


Recommended Literature


1.      Амелькин В.В. Дифференциальные уравнения, Минск, БГУ, 2012.

2.      Бибиков Ю.Н. Курс обыкновенных дифференциальных уравнений. Москва: «Высшая школа», 1991.

3.      Федорюк М.В. Обыкновенных дифференциальные уравнения. СПб.; Издательство «Лань», 2003.

4.      Филиппов А.Ф. Сборник задач по дифференциальным уравнениям. Москва «Наука», 1992.


1.      Богданов Ю.С., Мазаник С.А., Сыроид Ю.Б. Курс дифференциальных уравнений. Минск: «Университетское», 1996.

2.      Еругин Н.П. Книга для чтения по общему курсу дифференциальных уравнений. Минск: «Наук»,1972.

3.      Степанов В.В. Курс дифференциальных уравнений. Москва: Физматгиз, 1959.



Teaching Methods

Lecture with practical exercises, using elements of distance learning and electronic materials.


Teaching language



Requirements, current control

Test papers. Exam score consist of the current mark (40%) and the oral exam mark (60%).


Method of certification

Credit, Exam